Take a Square II

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

You may wish to look at Take a Square before trying this problem.
Diagonal and fold to midpoint
Diagonal and fold to one eighth
You can divide the long diagonal of a square into different fractions by folding.
  • In the first image the second fold joins a corner of the square to the midpoint of the opposite side.
  • In the second image the second fold joins a corner of the square to a point $\frac{1}{8}$ of the way along the opposite side.

This problem is about the fractions of the long diagonal of a square which you can construct in this way.

To start with, we shall only consider points on the side of the square which can easily be found by folding. That is, $\frac{1}{2}$s or $\frac{1}{4}$s or $\frac{1}{8}$s and so on.
Investigate the fractions of the long diagonal of a square that can be created in the way described above. Here are some examples to think about:

Quarter1Quarter2Quarter3Quarter4...All quarters


Eighth1Eighth2Eighth3Eighth4...All eighths

Can you extend the findings and make generalisations?
Can you justify your generalisations?

What about starting with fractions of the side of the square that are not so easily found by folding?

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Published June 2009.